It's a full service, CAS, IFR airport sensibly close to Shoreham and on the route from Bournemouth. It is used as an alternate from Bournemouth for IR tests, which was the main south coast IR test location until the recent changes. Any examiner is likely to be comfortable with Southampton as an alternate.
Lydd looks a good possibility also, but it is further, off-route, outside CAS, no radar and (perhaps most critically) the DHs and RVRs are minima are much higher (ie. 480' and 1500m). Southampton a better bet, given the relatively high minima at Shoreham.
I can't "prove" it "must" be Southampton. Lydd would be fine too if the TAF were suitable. It's obviously not an actual item someone would fail on.
brgds
421C